If A= [ array ll 1 & 2 2 & 3 array ] and A^2-k A-I₂=O , where I₂ is the 2 2 identity matrix, then what is the value of k ?

Step-by-step solution

aligned & A= [ array ll 1 & 2 2 & 3 array ] & A^2= [ array ll 1 & 2 2 & 3 array ] [ array cc 1 & 2 2 & 3 array ]= [ array cc 1+4 & 2+6 2+6 & 4+9 array ]= [ array cc 5 & 8 8 & 13 array ] & A ^2- kA - I ₂= [ array cc 5 & 8 8 & 13 array ]- [ array cc k & 2 k 2 k & 3 k array ]- [ array cc 1 & 0 0 & 1 array ] & = [ array cc 5- k -1 & 8-2 k -0 8-2 k -0 & 13-3 k -1 array ] & = [ array cc 4- k & 8-2 k 8-2 k & 12-3 k array ] aligned aligned & Given, A ^2- kA - I ₂=0 & 4- k =0 k =4 aligned